Documentation

Mathlib.Data.Finset.Erase

Erasing an element from a finite set #

Main declarations #

Tags #

finite sets, finset

erase #

source
def Finset.erase {α : Type u_1} [DecidableEq α] (s : Finset α) (a : α) :
Finset α

erase s a is the set s - {a}, that is, the elements of s which are not equal to a.

Equations
  • s.erase a = { val := s.val.erase a, nodup := ⋯ }
Instances For
    source
    @[simp]
    theorem Finset.erase_val {α : Type u_1} [DecidableEq α] (s : Finset α) (a : α) :
    (s.erase a).val = s.val.erase a
    source
    @[simp]
    theorem Finset.mem_erase {α : Type u_1} [DecidableEq α] {a b : α} {s : Finset α} :
    a ∈ s.erase b ↔ a ≠ b ∧ a ∈ s
    source
    theorem Finset.not_mem_erase {α : Type u_1} [DecidableEq α] (a : α) (s : Finset α) :
    a ∉ s.erase a
    source
    theorem Finset.ne_of_mem_erase {α : Type u_1} [DecidableEq α] {s : Finset α} {a b : α} :
    b ∈ s.erase a → b ≠ a
    source
    theorem Finset.mem_of_mem_erase {α : Type u_1} [DecidableEq α] {s : Finset α} {a b : α} :
    b ∈ s.erase a → b ∈ s
    source
    theorem Finset.mem_erase_of_ne_of_mem {α : Type u_1} [DecidableEq α] {s : Finset α} {a b : α} :
    a ≠ b → a ∈ s → a ∈ s.erase b
    source
    theorem Finset.eq_of_mem_of_not_mem_erase {α : Type u_1} [DecidableEq α] {s : Finset α} {a b : α} (hs : b ∈ s) (hsa : b ∉ s.erase a) :
    b = a

    An element of s that is not an element of erase s a must bea.

    source
    @[simp]
    theorem Finset.erase_eq_of_not_mem {α : Type u_1} [DecidableEq α] {a : α} {s : Finset α} (h : a ∉ s) :
    s.erase a = s
    source
    @[simp]
    theorem Finset.erase_eq_self {α : Type u_1} [DecidableEq α] {s : Finset α} {a : α} :
    s.erase a = s ↔ a ∉ s
    source
    theorem Finset.erase_ne_self {α : Type u_1} [DecidableEq α] {s : Finset α} {a : α} :
    s.erase a ≠ s ↔ a ∈ s
    source
    theorem Finset.erase_subset_erase {α : Type u_1} [DecidableEq α] (a : α) {s t : Finset α} (h : s ⊆ t) :
    s.erase a ⊆ t.erase a
    source
    theorem Finset.erase_subset {α : Type u_1} [DecidableEq α] (a : α) (s : Finset α) :
    s.erase a ⊆ s
    source
    theorem Finset.subset_erase {α : Type u_1} [DecidableEq α] {a : α} {s t : Finset α} :
    s ⊆ t.erase a ↔ s ⊆ t ∧ a ∉ s
    source
    @[simp]
    theorem Finset.coe_erase {α : Type u_1} [DecidableEq α] (a : α) (s : Finset α) :
    ↑(s.erase a) = ↑s \ {a}
    source
    theorem Finset.erase_idem {α : Type u_1} [DecidableEq α] {a : α} {s : Finset α} :
    (s.erase a).erase a = s.erase a
    source
    theorem Finset.erase_right_comm {α : Type u_1} [DecidableEq α] {a b : α} {s : Finset α} :
    (s.erase a).erase b = (s.erase b).erase a
    source
    theorem Finset.erase_inj {α : Type u_1} [DecidableEq α] {x y : α} (s : Finset α) (hx : x ∈ s) :
    s.erase x = s.erase y ↔ x = y
    source
    theorem Finset.erase_injOn {α : Type u_1} [DecidableEq α] (s : Finset α) :
    Set.InjOn s.erase ↑s